
diff -r c6ab12960403 -r 0c681fbb7b85 text/ncat.tex
--- a/text/ncat.tex	Tue Jun 21 12:05:16 2011 -0700
+++ b/text/ncat.tex	Tue Jun 21 18:10:31 2011 -0700
@@ -689,6 +689,8 @@
 Furthermore, if $q$ is any decomposition of $X$, then we can take the vertex of $\vcone(P)$ to be $q$ up to a small perturbation.
 \end{axiom}
 
+\nn{maybe also say that any splitting of $\bd c$ can be extended to a splitting of $c$}
+
 It is easy to see that this axiom holds in our two motivating examples, 
 using standard facts about transversality and general position.
 One starts with $q$, perturbs it so that it is in general position with respect to $c$ (in the case of string diagrams)
@@ -696,7 +698,7 @@
 and the perturbed $q$.
 These common refinements form the middle ($P\times \{0\}$ above) part of $\vcone(P)$.
 
-We note two simple special cases of axiom \ref{axiom:vcones}.
+We note two simple special cases of Axiom \ref{axiom:vcones}.
 If $P$ is the empty poset, then $\vcone(P)$ consists of only the vertex, and the axiom says that any morphism $c$
 can be split along any decomposition of $X$, after a small perturbation.
 If $P$ is the disjoint union of two points, then $\vcone(P)$ looks like a letter W, and the axiom implies that the
@@ -885,7 +887,7 @@
 In the $n$-category axioms above we have intermingled data and properties for expository reasons.
 Here's a summary of the definition which segregates the data from the properties.
 
-An $n$-category consists of the following data:
+An $n$-category consists of the following data: \nn{need to revise this list}
 \begin{itemize}
 \item functors $\cC_k$ from $k$-balls to sets, $0\le k\le n$ (Axiom \ref{axiom:morphisms});
 \item boundary natural transformations $\cC_k \to \cl{\cC}_{k-1} \circ \bd$ (Axiom \ref{nca-boundary});




